protein docking by the underestimation of free energy

Protein Docking by the Underestimation of Free EnergyFunnels in the Space of Encounter ComplexesYang Shen1,2, Ioannis Ch. Paschalidis2,3,4, Pirooz Vakili2,3,5, Sandor Vajda1*

1 BioMolecular Engineering Research Center, Department of Biomedical Engineering, Boston University, Boston, Massachusetts, United States of America, 2 Center for

Information and Systems Engineering, Boston University, Boston, Massachusetts, United States of America, 3 Division of Systems Engineering, Boston University, Boston,

Massachusetts, United States of America, 4 Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts, United States of America,

5 Department of Mechanical Engineering, Boston University, Boston, Massachusetts, United States of America

Abstract

Similarly to protein folding, the association of two proteins is driven by a free energy funnel, determined by favorableinteractions in some neighborhood of the native state. We describe a docking method based on stochastic globalminimization of funnel-shaped energy functions in the space of rigid body motions (SE(3)) while accounting for flexibility ofthe interface side chains. The method, called semi-definite programming-based underestimation (SDU), employs a generalquadratic function to underestimate a set of local energy minima and uses the resulting underestimator to bias furthersampling. While SDU effectively minimizes functions with funnel-shaped basins, its application to docking in the rotationaland translational space SE(3) is not straightforward due to the geometry of that space. We introduce a strategy that usesseparate independent variables for side-chain optimization, center-to-center distance of the two proteins, and five angulardescriptors of the relative orientations of the molecules. The removal of the center-to-center distance turns out to vastlyimprove the efficiency of the search, because the five-dimensional space now exhibits a well-behaved energy surfacesuitable for underestimation. This algorithm explores the free energy surface spanned by encounter complexes thatcorrespond to local free energy minima and shows similarity to the model of macromolecular association that proceedsthrough a series of collisions. Results for standard protein docking benchmarks establish that in this space the free energylandscape is a funnel in a reasonably broad neighborhood of the native state and that the SDU strategy can generatedocking predictions with less than 5 A ligand interface Ca root-mean-square deviation while achieving an approximately 20-fold efficiency gain compared to Monte Carlo methods.

Citation: Shen Y, Paschalidis ICh, Vakili P, Vajda S (2008) Protein Docking by the Underestimation of Free Energy Funnels in the Space of EncounterComplexes. PLoS Comput Biol 4(10): e1000191. doi:10.1371/journal.pcbi.1000191

Editor: Matthew P. Jacobson, University of California San Francisco, United States of America

Received June 2, 2008; Accepted August 22, 2008; Published October 10, 2008

Copyright: � 2008 Shen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This research was supported by National Institutes of Health/National Institute of General Medical Sciences grants R21-GM079396 and R01-GM061867.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

Introduction

Genomewide proteomics studies, primarily yeast two-hybrid

assays [1,2] and high-throughput mass spectrometry [3,4], provide

a growing list of putative protein–protein interactions, and

demonstrate that most if not all proteins have interacting partners

in the cell. Elucidating the atomic details of these complexes

requires further biochemical and structural information. While the

most complete structural characterization of a complex is provided

by X-ray crystallography, protein–protein hetero-complexes

constitute less than 2% of protein structures in the Protein Data

Bank (PDB) [5], and their number increases at a slow rate. In fact,

many biologically important interactions occur in weak, transient

complexes that will not be amenable to direct experimental

analysis, even when both proteins can be isolated and their

structures determined. Thus, there is substantial need for

computational docking methods that can determine the structure

of a complex from the separately solved structures of two

component proteins.

Based on the thermodynamic hypothesis, at fixed temperature

and pressure the Gibbs free energy of the macromolecule-solvent

system reaches its global minimum at the native state of the

complex. Thus, docking requires a computationally feasible free

energy evaluation model and an effective minimization algorithm.

It is expected that docking methods can utilize the rich set of

modeling tools developed for predicting the structures of folded

proteins. It has been established over the last two decades that the

energy landscape of a foldable protein resembles a many-

dimensional funnel with a free energy gradient toward the native

structure [6–9]. A number of papers suggest that the landscape

theory also applies to protein–protein association [10–12]. The

size of the funnel is determined by the length scales of the long-

range electrostatic and hydrophobic interactions and the geometry

of the proteins, and hence the funnel is restricted to a

neighborhood of the native complex [13]. There is a free energy

gradient toward the native state, but the funnel is rough, giving rise

to many local minima [14] that correspond to encounter

complexes, some of which may be visited along a particular

association pathway [15,16].

While homology modeling approaches play an important role in

protein structure prediction, most current docking methods are

based on direct optimization, and attempt to find the global

minimum of a function approximating the free energy of the

complex. According to the results of CAPRI (Comparative

Assessment of PRotein Interactions), a community-wide experi-

ment devoted to protein–protein docking [17–20], the optimiza-

PLoS Computational Biology | www.ploscompbiol.org 1 October 2008 | Volume 4 | Issue 10 | e1000191

tion involves either the systematic sampling of the discretized

rotational/translational space using Fast Fourier Transforms

[21,22] or geometric hashing [23], or it relies on Monte Carlo

(or Monte Carlo minimization) algorithms [24,25]. Both optimi-

zation methods are generic, i.e., they do not rely on any

assumption about the specific shape of the energy function to be

minimized.

The use of special optimization methods that account for the

funnel-like shape of the free energy function offers two potential

advantages. First, being designed for minimizing funnel-like

functions, such algorithms can be more efficient than generic

approaches. Second, the success of such algorithms will be a

stringent test of how well the funnel assumption describes the binding

free energy landscape. This second point is particularly interesting,

because protein–protein association occurs in the six-dimensional

(6D) space of translations and rotations, at least for the classes of

proteins whose backbones remain essentially unchanged upon

association (e.g., many enzymes interacting with their inhibitors).

Although the association is accompanied by conformational

changes, these can be considered auxiliary, and the shape of the

funnel can be studied over the entire conformational space. In

contrast, the free energy of protein folding is defined in a

substantially higher-dimensional space, and hence funnels can be

generally studied only along some reaction coordinates [6–9].

Minimization by UnderestimationA minimization approach which is specific to funnel-like

functions can be based on the concept of underestimation. The

existence of a funnel implies that the free energy can be locally

underestimated by a convex function (Figure 1). The original free

energy function is extremely rugged with a huge number of local

minima even in a small region of conformational space. Yet its

convex underestimator is much smoother and still captures the

overall funnel-like landscape, which provides a handle to free

energy minimization. The quality of minimization through

underestimation depends on the choice of underestimator

functions, the way they are constructed and utilized to locate the

global minimum, as well as how structured the free energy funnels

are in conformational space. The Convex Global Underestimation

(CGU) method [26] employed canonical quadratic functions as

underestimators without any cross-terms. In that case the under-

estimator, based on a set of local minima, can be constructed by

solving a Linear Programming (LP) problem. Uniformly distrib-

uted samples in the neighborhood of the underestimator’s global

minimum were then used to bias further sampling. The process

was iterated with the set of local minima being updated, and the

search region being reduced until certain convergence criteria are

satisfied. CGU has been a very promising method with various

applications in molecular structure prediction, including protein

folding [27] and docking small molecules to proteins [28].

However, its restriction of using canonical quadratic functions

limits its success in some cases [29], since the principal axes of the

free energy surface are not necessarily aligned with the canonical

coordinates. The performance further deteriorates as the dimen-

sionality of the search space increases. We have used theoretical

analysis to show and simple test problems to demonstrate that this

restriction can lead to incorrect convergence [30].

Motivated by the potential advantages of underestimation and

practical shortcomings of the CGU algorithm, we have recently

extended the method using general quadratic underestimators and

introduced biased sampling guided by the underestimator [30,31].

Since the tightest underestimator in this class is obtained by solving

a semi-definite programming problem, the method is termed SDU

(Semi-Definite programming-based Underestimation). Semi-defi-

nite programming is computationally more demanding than the

linear programming (still solvable in polynomial-time though) used

in the CGU method. However, SDU typically requires fewer

iterations and substantially improves optimization performance.

The SDU method starts from a set of K local minima x1, …, xK

of a funnel-like function f(x): Rn?R within a given region B of the

search space. Throughout the course of the algorithm we maintain

a set L of local minima in the search region; initially

L~ x1, . . . , xK� �

. To capture the global funnel-like structure of

f(x) within B we construct a smooth (convex) quadratic function

U(x) = x9Qx+b9x+c, where Q is a positive semi-definite matrix,

b[Rn, c is a scalar, and prime denotes transpose, such that U(x)

underestimates f(x) at all local minima in L, i.e., U(xi)#f(xi) for all

i = {1, 2, …, K}. The tightest possible underestimator (with an L1

Figure 1. Funnel-like function and underestimator at a set oflocal minima indicated by small squares.doi:10.1371/journal.pcbi.1000191.g001

Author Summary

Protein–protein interactions play a central role in variousaspects of the structural and functional organization of thecell, and their elucidation is crucial for a better under-standing of processes such as metabolic control, signaltransduction, and gene regulation. Genomewide proteo-mics studies, primarily yeast two-hybrid assays, willprovide an increasing list of interacting proteins, but onlya small fraction of the potential complexes will beamenable to direct experimental analysis. Thus, it isimportant to develop computational docking methodsthat can elucidate the details of specific interactions at theatomic level. Protein–protein docking generally starts witha rigid body search that generates a large number ofdocked conformations with good shape, electrostatic, andchemical complementarity. The conformations are clus-tered to obtain a manageable number of models, but thecurrent methods are unable to select the most likelystructure among these models. Here we describe arefinement algorithm that, applied to the individualclusters, improves the quality of the models. The bettermodels are suitable for higher-accuracy energy calculation,thereby increasing the chances that near-native structurescan be identified, and thus the refinement increases thereliability of the entire docking algorithm.

Protein Docking by Underestimation


norm metric) can be found by solving a semi-definite program-

ming problem [30]. U(x) is a general convex quadratic function.

The underestimator U is used to guide further sampling. The

minimum of U, denoted by xP and referred to as the predictive

conformation, is in an energetically favorable region, and hence a new

conformation can be generated by local minimization starting from

xP. Additional conformations are obtained by local minimization

with randomly generated starting points such that points in the

vicinity of xP have a higher probability of being selected than points

further away. To that end, we simply sample within B using a density

function shaped as 2U. The set L is being updated by adding these

newly obtained conformations while removing unfavorable (i.e.,

higher energy) conformations, and the search area B is being

reduced to a neighborhood of xP. Using the updated conformations

in L we repeat the underestimation step and the whole process is

being iterated until a convergence criterion is met.

The SDU algorithm has a number of favorable properties when

applied to funnel-shaped functions. Theoretical analysis shows

[30] that xP converges in probability to the global minimum x* of

the funnel-like function f as the number of samples K grows. When

applied to test functions resembling the funnel-like free energy

functions, SDU has been shown outperforming CGU and a

simulated annealing algorithm which adaptively tunes its param-

eters, with much less required function evaluations and much

higher success rates [30].

Docking by Semi-Definite UnderestimationThe main goal of this paper is to develop and test docking

methods that use the SDU algorithm in order to find the global

minimum of a funnel-like function approximating the free energy

over regions of the conformational space. Over the last few years

we have developed a multistage docking method that starts with

rigid body search based on Fast Fourier Transform (FFT), selects

and clusters 1000 to 2000 low energy docked structures, and

retains the 10 to 30 largest clusters for further processing [32]. The

conformational space is decomposed into separate regions defined

by the clusters, where free energy attraction basins are believed to

exist and the free energy landscape is assumed to be funnel-like.

Then SDU is called upon to locate the global minimum within

each region, by utilizing such funnel-like behavior.

As will be shown, in spite of the success of SDU as an optimizer

for functions with funnel-shaped basins, its application to docking

turned out to be far from straightforward. Although the method

yields meaningful moves in either translational or rotational

subspaces [30], minimization in the full 6D space of rigid body

motions poses a challenge. This difficulty is well known in the

robotics literature [33]. The Euclidean group SE(3), which is the

semidirect product of R3 (translations) and the special orthogonal

Euclidean group SO(3) (rotations), is a nonlinear manifold and its

parametrization is critical to any optimization procedure [33]. In

particular, we will show that the funnel-like shape of the free

energy surface is affected by the parameterization of the search

space, and that underestimation tends to fail unless appropriate

parameters are selected.

We will describe two implementations of the SDU method: (i)

SDU1, a cyclic coordinate descent strategy where rotational and

translational moves alternate; and (ii) SDU2, a 5D strategy in

which the distance of the two proteins is separately optimized and

the SDU-driven search is performed in a lower dimensional space

defined by 5 angular coordinates. Since the most energetically

favorable distances occur when the two proteins are in contact but

do not overlap, this strategy explores the free energy surface

spanned by encounter complexes [15]. Results will show that the

methods discover broad energy funnels, generate high quality

docking predictions, and produce a substantial efficiency gain

compared to Monte Carlo methods.

The comparison with Monte Carlo methods is based on our

earlier preliminary work [34] where we have tested a method

similar to SDU1 against a 10-protein set using reduced Go-type

potentials. Even with these relatively smooth potentials, underes-

timation in the full 6D space has not been effective, and we had to

rely on optimizing in each (rotational or translational) subspace

separately. In the present paper, we show that the SDU1 strategy

is effective against a larger benchmark set using much more

refined energy potentials. However, our main contribution is the

introduction of SDU2 which is more effective and improves

efficiency by a factor of 10 compared to SDU1. In addition, SDU2

provides interesting biophysical insights with its resemblance to

docking by repeated micro-collisions.

Results/Discussion

Development of SDU-Based Docking AlgorithmsTo define the docking problem we fix the position and orientation

of the first (receptor) protein. The six-dimensional vector x specifies

the position and orientation of the second (ligand) protein. The

variables s account for the side chain conformations in both proteins,

and DG(x,s) denotes the free energy function. The minimization of

DG(x,s) with respect to s is restricted to the side chains in the

interface and is carried out by local methods. We define

DG*(x) =DG(x,s*), where s* is an optimal solution; then the protein

docking problem is finding the lowest minimum of DG*(x) in the

region defined by the cluster, where x belongs to the space of rigid

body motions, i.e., the Euclidean group SE(3). As described before,

the parameterization of SE(3) is critical for optimization purposes,

and hence we first describe our results concerning the parameter-

ization of the search space.

The space SE(3) is the semidirect product of R3 (translations)

and SO(3) (rotations). The rotation group SO(3) is a 3-dimensional

Lie group consisting of rotation matrices, i.e, SO 3ð Þ~R R[R3|3��

, RR9 = I, det(R) = 1. The Lie algebra of SO(3),

denoted by so(3), may be represented by the real skew-symmetric

matrices

V~

0 {v3 v2

v3 0 {v1

{v2 v1 0

264

375,

where v~ v1,v2,v3ð Þ[R3. It is well-known that the one-

parameter subgroups of SO(3), i.e., R tð Þ~eVt V[so 3ð Þ, t[Rð Þ are

geodesics [33,35,36], i.e., the shortest paths between two points on

SO(3). Moreover, for R0, R1MSO(3), V= log(R09R1)Mso(3) and the

distance between R0 and R1 can be defined by r(R0,R1) =

Ilog(R09R1)I = Iv12v0I. This distance is a natural Riemannian

metric on SO(3), i.e., it is bi-invariant with respect to the actions of

the group (rotations). The exponential map from so(3) to SO(3)

defined by V R R0eV is a local diffeomorphism, i.e., there exist an

open neighborhood of 0Mso(3) and an open neighborhood of

R0MSO(3) and an invertible, surjective, and smooth map from one

neighborhood to the other whose inverse is also smooth. The local

diffeomorphism induces a coordinate chart in a neighborhood of

R0 that is known as exponential coordinate system. Given the

definition of so(3) this coordinate system can be parameterized by

v~ v1,v2,v3ð Þ[R3.

Given the favorable properties of SO(3) (more generally SO(n))

such as the existence of a natural bi-invariant metric and, in

particular, the simplicity of determining the geodesics of the



manifold, the standard gradient based optimization algorithms on

Rn can be generalized for optimization on SO(n) [35]. However,

generalization to the entire SE(3) is more difficult [33,36].

Although SE(3) is also Lie group, its one-parameter subgroups

are no longer geodesics. Moreover, SE(3) does not admit a natural

(bi-invariant) Riemannian metric while its subspaces SO(3) and R3

do [33,36]. Attempting to implement SDU to SE(3) by

simultaneous translational and rotational optimization, we have

generally failed to construct useful underestimators. We have,

however, found two strategies that were able to overcome this

problem.

SDU1: Cyclic coordinate descent. A natural search strategy

in SE(3) is to alternate between optimizing the free energy in SO(3)

and in R3 by a series of rotational and translational adjustments.

As will be further discussed, the major disadvantage of this

approach is that samples in one subspace cannot be reused in the

other subspace, resulting in inefficient search.

SDU2: 5D search in the space of encounter

complexes. The distance between the two proteins is

separately optimized along the center-to-center vector, and the

SDU search space is reduced to S26SO(3), where S2 denotes the

surface of the unit sphere in R3. The receptor is fixed, with its

center of mass placed at the origin of the coordinate system, and

y[R3 denotes the position of the center of mass of the ligand. In

spherical coordinates y can be represented by (r,h,Q), where

r = IyI, h is the azimuth angle between the projection of y on the

xy-plane and the x-axis (longitude, 0#h,2p), and Q is the zenith

angle between the z-axis and the vector y (colatitude, 0#Q#p).

The corresponding exponential coordinates are s = (2sin h?Q, cos

h?Q). The rotation of the ligand is described by the exponential

coordinates v[R3 in SO(3).

In this coordinate system the free energy function is DG(r,s,v,s),

where s describes the side chain conformations. Since s and r,

respectively, are determined by local minimization and by a line

search along the vectors connecting the centers of mass, by SDU

we minimize the function

DG� s,vð Þ~ minr,s

DG r,s,v,sð Þ,

in the (s,v)-space. Thus, the SDU2 algorithm uses separate

independent variables for side-chain optimization, center-to-center

distance of the two proteins, and five angular descriptors of the

relative orientations of the molecules. As will be shown, the

removal of the center-to-center distance turns out to vastly

improve the efficiency of the search, since the 5-dimensional space

now exhibits a well-behaved energy surface suitable for underes-

timation. In addition, successive underestimators obtained during

the course of the SDU2 algorithm can reuse the local minima

obtained in the earlier steps, thereby reducing the number of

required function evaluations. Moreover, SDU2 can use all

conformations contained in the cluster to be refined, while

SDU1 may use very few (or none) of these points as they may

not lie in the subspaces explored. In fact, we have tested SDU1 for

a 10-protein set with reduced energy potentials and compared its

performance with a standard Monte Carlo method in [34]. SDU1

showed a modest speed-up factor of 2 compared with the Monte

Carlo method, partly due to the issues mentioned above.

It is important that for any rotational state the energy is separately

minimized along the vector connecting the centers of the two

molecules. Since the lowest energy is generally attained at a distance

that eliminates all atomic overlaps but retains some of the favorable

van der Waals interactions, after minimization the two proteins are

in contact with each other. Based on this property, the SDU2

algorithm essentially samples encounter complexes [15], resulting in

meaningful energy values and efficient sampling. Simple arguments

show that this parameterization is more natural than the sampling in

the space of translations and rotations. In fact, the SDU2 strategy

shows strong similarity to the model of macromolecular association

in which translational diffusion brings the two proteins to a collision.

Unless the enthalpy change of favorable interactions compensates for

the free energy increase due to the loss of entropy, the proteins

separate, rotate, and collide again. Thus, the conformational search

proceeds in a series of ‘‘micro-collisions’’, each resulting in an

encounter complex [37].

Test Results for SDU-Based Docking AlgorithmsThe SDU1 and SDU2 algorithms were tested on the protein

pairs given in protein docking benchmark sets [38,39] that contain

enzyme–inhibitor, antigen–antibody, and ‘‘other’’ types of com-

plexes, using the independently determined (unbound) structures

of the component proteins. The algorithms were used to refine the

top (most populated) clusters of docked structures generated by the

rigid body docking program PIPER [22]. We consider the 10

largest clusters for enzyme–inhibitor complexes, and 30 clusters for

antigen–antibody and ‘‘other’’ complexes. Table 1 shows results

both from the rigid body docking and the SDU-based refinement

procedures for all three types of complexes, each defined by its

Protein Data Bank (PDB) code [5] in Column 1 of Table 1. We

emphasize that for most complexes we docked the unbound

(separately crystallized) protein structures rather than their bound

conformations obtained by separating the complex. The excep-

tions are mostly a few antigen–antibody complexes for which no

separate antibody structures were available and hence were taken

from the complex. However, even in these cases we used the

separately crystallized structure of the antigen as given in the

benchmark sets [38,39]. PDB codes for these ‘‘semi-bound’’

complexes are shown in bold italic fonts in Table 1.

Columns 2–4 of Table 1 describe the docked complex structures

generated by the rigid body docking [22] before any refinement.

As described in the Methods, the PIPER docking program

evaluates the energy for billions of docked conformations. We

retain the 1000 best scoring structures, and cluster them using the

pairwise RMSD as the distance measure and an optimally selected

clustering radius. We have observed that the near-native structures

tend to be in one of the largest clusters, and hence rank the clusters

on the basis of their size. In fact, calculated for the rigid protein

structures, the energy function is approximate, and better

discrimination of the native structures can be achieved by focusing

on the large clusters. The properties shown in Table 1 are the

number of near-native conformations (or ‘‘hits’’ with less than 10

A ligand interface Ca RMSD) among the 1000 best scoring

structures retained from the PIPER results, the Root Mean Square

deviation (RMSD) between the native ligand structure and the

docked structure at the center of the first cluster that includes a

near-native conformation, and the rank of the particular cluster

based on cluster size. Results are fairly good for enzyme–inhibitor

complexes, as the 1000 structures, on the average, include over

200 hits, and the average RMSD is less than 5 A between the

native structure and the center of one of the three largest clusters.

Although PIPER yields almost 200 hits for the ‘‘other’’ types of

complexes, discrimination by cluster size is more difficult, and

retaining the top 6 clusters results in 6.16 A average RMSD. For

antibody-antigen complexes PIPER generates much fewer hits,

and on the average we have to retain 15 clusters to have a near-

native structure in them.

The next 6 columns in Table 1 show the results for the SDU1

and SDU2 algorithms, in each case listing the RMSD for the



Table 1. Docking and refinement results.

Complexa PIPER SDU1 SDU2 ZDOCK+RDOCK

Hits RMSD Rank RMSD Rank n RMSD Rank n RMSD Rank

Enzyme–Inhibitor Complexes

1ACB 632 6.11 1 0.86 2 1425 2.78 3 93.8 5.66 1

1AVW 81 3.87 4 3.31 1 1381 3.54 2 90.0 2.50 1

1AVX 193 3.27 1 2.57 1 1393 4.18 1 91.2 2.62 39

1BRC 375 8.60 2 9.97 2 1431 4.50 2 98.5 8.92 1

1BVN 443 6.65 1 5.95 2 1498 1.11 1 81.7 1.78 8

1CGI 477 8.54 1 7.77 3 1425 2.40 3 100.0 9.03 16

1CHO 510 1.10 1 2.08 1 1475 2.64 2 91.4 5.25 1

1CSE 45 2.15 5 3.03 2 1340 5.99 2 97.3 2.83 6

1DFJ 30 6.59 4 3.80 2 1378 7.20 1 97.6 5.46 1

1E6E 55 6.88 9 4.85 1 1506 5.12 3 84.7 5.39 3

1EAW 114 2.02 3 3.03 2 1442 3.96 3 97.1 3.21 30

1MAH 171 1.80 3 1.37 1 1413 1.44 3 91.2 2.48 6

1PPE 605 4.24 1 2.42 1 1435 2.34 1 96.7 2.58 1

1STF 35 2.12 8 1.31 1 1303 0.80 1 92.7 1.70 4

1TGS 365 4.01 1 5.37 1 1343 6.02 1 98.0 6.18 158

1TMQ 25 3.03 9 1.01 2 1420 3.09 4 89.3 4.99 13

1UDI 217 4.37 1 1.73 2 1401 2.79 1 100.8 2.42 18

1UGH 97 4.75 3 4.86 1 1386 4.75 1 92.0 3.63 1

2MTA 161 5.30 1 6.47 3 1478 5.25 6 86.7 6.54 428

2PTC 322 7.49 1 7.37 4 1306 7.88 3 98.0 4.72 68

2SIC 76 7.76 4 1.93 1 1368 2.62 1 94.5 1.99 2

2SNI 103 8.56 2 8.71 1 1386 9.33 1 99.3 7.08 9

2TEC 235 1.89 1 1.56 1 1375 1.69 1 91.7 4.85 1

4HTC 73 5.21 7 4.82 3 1436 1.79 1 85.8 1.41 2

7CEI 178 7.68 1 6.61 3 1345 6.62 5 97.5 6.15 1

Average 224.7 4.96 3.0 4.11 1.76 1403 3.99 2.12 93.5 4.37 32.8

Antigen–Antibody Complexes

1AHW 131 5.94 3 5.16 13 1163 2.83 4 98.7 3.08 2

1BVK 41 5.66 28 9.84 8 1249 5.80 29 96.0 6.34 435

1EO8 25 6.15 29 {11.60} 30 1194 2.50 6 97.3 N/A N/A

1FBI 32 6.49 20 9.47 14 1212 3.43 17 92.7 7.97 266

1IAI 35 7.11 4 7.97 2 1210 6.20 4 96.7 7.90 212

1MEL 112 5.90 2 6.34 7 1239 2.65 4 100.0 3.52 103

1MLC 16 5.86 26 8.96 22 1242 7.57 19 93.3 9.50 47

1QFU 54 4.20 3 7.32 24 1091 2.54 22 90.0 2.97 96

1WEJ 26 3.65 19 5.45 19 1257 1.89 1 90.0 2.41 6

2JEL 136 6.74 4 5.40 6 1246 4.74 10 99.3 6.84 142

2VIR 14 8.90 25 8.50 28 1223 {11.12} 26 95.3 N/A N/A

Average 56.6 6.05 14.8 7.82 15.7 1211 4.66 12.7 93.4 5.61 145.4

Other Complexes

1A0O 193 6.01 2 6.91 1 1230 8.45 1 89.3 3.27 50

1ATN 272 6.57 1 6.38 4 1046 0.87 2 83.3 2.72 16

1GLA 396 8.38 1 5.65 3 1366 3.03 4 84.3 2.29 28

1MDA 60 7.59 15 {12.00} 13 1179 8.19 22 92.0 N/A N/A

1SPB 252 3.41 3 1.80 1 1162 1.40 1 90.7 1.31 1

1WQ1 143 8.44 3 4.99 9 1164 10.00 1 93.3 4.88 656

2BTF 140 0.81 2 0.91 1 1165 1.68 1 86.0 2.50 1



lowest energy conformation in the first cluster that includes a near-

native structure, the rank of the cluster, and the average number n

of function evaluations. It is important to note that after the SDU

refinement the clusters are ranked based on the energies of the

SDU solutions rather than on cluster size. For comparison we also

generated 2000 conformations (based on the original protocol) for

each complex using version 2.3 of the rigid body docking program

ZDOCK [21], and refined the structures using RDOCK [40],

which performs local energy minimization. The last two columns

of Table 1 show the RMSD of the first near-native structure found

and its rank based on the RDOCK energy function.

For enzyme–inhibitor complexes both SDU1 and SDU2 give

excellent results. On the average, the RMSD is reduced by almost

1 A and the average rank is around 2. As the overall performance

measure of the refinement we consider the number of complexes

that have at least one prediction with less than 5 A ligand interface

Ca RMSD from the native structure in the top 5 clusters. Such

predictions would be termed ‘‘5 A models’’ later for simplicity. As

noted, the clusters of PIPER-generated structures are ranked

based on their size (i.e., the number of structures). However, after

the refinement the clusters are ranked based on the energy of their

lowest energy structures. Among the PIPER-generated structures

the top 5 clusters include less than 5 A ligand interface RMSD

predictions for only 11 of the 25 enzyme–inhibitor complexes. As

shown in Table 1, both SDU1 and SDU2 increase this number to

17, i.e., a more than 50% improvement relative to the PIPER

results. Notice that the 5 lowest energy predictions obtained by the

ZDOCK/RDOCK procedure include 5 A models only for 7

complexes. However, from the ZDOCK/RDOCK runs we retain

low energy models, whereas the results provided by SDU1 and

SDU2 are low energy clusters, and ranking clusters rather than

individual structures generally removes some false positives, i.e.,

conformations that have low energy but are far from the native.

It is well known that antigen–antibody and ‘‘other’’ complexes

are more difficult to predict than enzyme–inhibitor complexes

[20]. For antibody-antigen pairs only the refinement by SDU2

improves the PIPER results. Although both the RMSD and the

average rank of the first near-native cluster are reduced, at 12.7

the latter remains high (Table 1). Similarly, for the complexes in

the ‘‘other’’ category only SDU2 improves both the RMSD and

the rank. Even SDU2 yields only a total of seven 5 A models in the

top five clusters for antibody–antigen and ‘‘other’’ types of

complexes. This result is somewhat disappointing, but note that

docking by ZDOCK and refining by RDOCK leads to 5 A models

only for three complexes, and no near-native solution is found in

four cases.

Calculated Free Energy SurfacesFigure 2 shows the RMSD vs. calculated free energy for the 25

enzyme–inhibitor complexes. Again we show PDB codes for the

‘‘semi-bound’’ complexes in bold italic fonts. Each point represents

a structure sampled in the process of refining the 10 largest clusters

using the SDU2 algorithm. The encircled blue asterisk indicates

the native structure and the first hit is shown as a red square. In

spite of the one-dimensional representation of the energy function

defined in the 6D rotational and translational space, the figure

demonstrates the multi-funnel behavior over a relatively broad

region (within 20 A RMSD) of the native state. For most

complexes the figure shows a well defined deep funnel within 5 A

RMSD from the native structure. According to Table 1, for 12 of

the 25 complexes (including 5 ‘‘semi-bound’’ cases), this funnel is

deepest among the 10 clusters sampled. For example, for 4HTC

the near-native cluster is the 7th largest, but it is energetically the

most favorable after refinement. For the remaining 13 complexes,

clusters farther from the native structure yield the deepest funnels,

resulting in false positive predictions. One of the worst behaviors

can be observed for 2PTC with the first near-native cluster ranked

4, and Figure 2 shows a number of deep non-native funnels.

However, on the average, one of the 3 deepest funnels is near-

native (Table 1). This also shows the power of heuristic

combination of entropic and enthalpic measurement, i.e., cluster

size as a filter and refined cluster depth as ranking parameter.

Figure 3 shows the RMSD vs. calculated free energy for the

antigen–antibody and ‘‘other’’ complexes, sampled in the

refinement of the 30 largest clusters by SDU2. All but 1SPB and

2BTF have multiple funnels, and the funnels are further from the

native state than for enzyme–inhibitor pairs, demonstrating the

well-known difficulty of estimating free energy using simple

models, particularly for antigen–antibody complexes. Adjusting

the conformation of interface side chains only by local minimi-

zation and keeping the backbone rigid also limits the accuracy of

free energy calculation.

The Effect of Space Selection on Search EfficiencyAs shown in Table 1, the SDU2 strategy is much more efficient

than SDU1, and also provides substantially better results for the

more difficult problem of docking antigen–antibody and ‘‘other’’

complexes. These differences are demonstrated in Figure 4A and

4B that show, for the near-native cluster of the complex 4HTC,

the conformations sampled and minimized by each algorithm.

The horizontal and vertical axes, respectively, represent transla-

tional and rotational distances between each sampled ligand

conformation and the one in the native structure, the latter placed

at the origin of this coordinate system. The rotational distance

here is defined as the length of the minimum geodesic (i.e., the

minimum rotation in radian) between two rotations [36]. As

discussed earlier, this distance between R0 and R1 can be defined

by r(R0,R1) = Ilog(R09R1)I = Iv12v0I. The points are color-

coded according to their energies, from low (blue) to high (red)

energy structures.

As shown in Figure 4A, the separate treatment of translational

and rotational subspaces in the SDU1 algorithm is highly

inefficient, resulting in the sampling of many conformations with

relatively high energies. Although all energy values shown are

Complexa PIPER SDU1 SDU2 ZDOCK+RDOCK

Hits RMSD Rank RMSD Rank n RMSD Rank n RMSD Rank

2PCC 59 8.12 16 7.78 26 1168 9.96 1 93.6 N/A N/A

Average 189.3 6.16 5.4 5.80 7.2 1185 5.44 4.1 89.1 5.62 125.3

aPDB codes for the bound-unbound docking problems are shown in italics. All others are unbound-unbound cases.doi:10.1371/journal.pcbi.1000191.t001

Table 1.cont.



obtained by local minimization, the latter is unable to reduce the

energy if the two proteins are too far from each other, which

frequently occurs with the SDU1 algorithm. In addition, as

indicated by the parallel lines, very similar translational regions are

re-sampled at slightly different rotational coordinates, and vice

versa. In contrast, as shown in Figure 4B, the SDU2 algorithm

smoothly and efficiently descends toward the bottom of the free

energy funnel. Based on the ranges of sampled free energy values,

SDU2 sampled much lower energy regions compared to SDU1.

Since the search is restricted to biophysically meaningful

encounter complexes, the more consistent energy values facilitate

the construction of better underestimators during the search.

According to Figure 4B, for SDU2 there is a clear trend that

structures closer to the native complex generally have lower

energies, resulting in a deep and broad free energy funnel in this

space. The existence of such free energy funnels is much less

obvious, even in each separate subspace, when sampled by the

SDU1 algorithm (Figure 4A).

In test docking problems even the slower SDU1 algorithm

outperformed a standard Monte Carlo method by reducing the

number of function evaluations by a factor of two [34]. As shown

in Table 1, SDU2 further reduces the computational costs by a

factor of 11 to 15, depending on the type of complex. Since most of

the computational time is spent in energy evaluations, the

computational gain of SDU2 over SDU1 is more than a factor

of 10, and we estimate that SDU2 achieves more than 20-fold

efficiency gain compared to Monte Carlo methods. We tested our

algorithms on a 128-node biowulf cluster (IBM eServer6Series).

Each node contains dual 1 GHz PIII processors with 2 GB

memory. A typical refinement by SDU2 for each PIPER-

generated cluster would take 2 to 6 processor-hours. The running

time varies with the protein complex size (especially the interface

size) and the number of iterations before convergence. Notice that

on the average SDU2 samples only about 100 encounter structures

for each cluster. No particular efforts have been made to

accelerate either the interface side-chain search or the line search

to determine the center-to-center distance. The CPU times can be

compared to those reported for a server [41] based on the

RosettaDock algorithm [42]. The server performs 1000 indepen-

dent Monte Carlo simulations within 30 A Ca RMSD of a starting

structure, as described in the RosettaDock protocol [42]. A typical

run requires about 65 processor-hours.

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Figure 2. RMSD vs. energy plots for SDU2 sampled structures in the near-native funnel for enzyme–inhibitor complexes. The nativestructure is indicated by a blue circled asterisk and the SDU2 prediction by a red square.doi:10.1371/journal.pcbi.1000191.g002



Repeatability of the SDU-Based Docking AlgorithmSDU based docking algorithms are stochastic in nature.

Although we gave a theoretical guarantee of probabilistic

convergence to the global minimum of funnel-like functions under

some fairly general conditions [30], practical protein docking

problems do not necessarily satisfy all these conditions. To check

the variations in our results we have repeatedly run SDU2 to

refine the near-native cluster from PIPER (columns 3 and 4 in

Table 1) for 10 randomly selected complexes, including 4 enzyme–

inhibitor, 4 antigen–antibody and 2 ‘‘other’’ complexes. SDU2

was run 10 times independently for each complex with the same

set of parameters. The results (ligand interface Ca RMSD versus

corresponding free energy values) are shown in Figure 5, where

black circles represent the rigid body predictions from PIPER before

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Figure 3. RMSD vs. energy plots for SDU2 sampled structures in the near-native funnel for antigen–antibody or other complexes.The native structure is indicated by a blue circled asterisk and the SDU2 prediction by a red square.doi:10.1371/journal.pcbi.1000191.g003

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Figure 4. SDU1 (A) and SDU2 (B) trajectories in the near-native cluster for 4HTC. Translational and rotational distances plotted in thehorizontal and vertical axes, are defined as Ir12r0I and Iv12v0I, respectively, where (r1,v1) denotes the position and orientation of the ligand and(r0,v0) are the corresponding values in the native state. The points are color-coded depending on their energy (in Kcal/mol as shown on the right),and the points in the squares show the best local minimum in each iteration.doi:10.1371/journal.pcbi.1000191.g004



refinement and blue asterisks represent the independent SDU2

predictions. Some of the latter overlap, resulting in less than 10

distinct solutions. SDU2 is able to lower the free energy values in all

cases and to improve RMSD in most of them. Table 2 shows the

mean (indicated by overline) and the standard deviation (indicated

by s) for both the ligand interface RMSD and the free energy for

each complex. Although the average standard deviation is relatively

large (about 4.4 Kcal/mol) for free energies, it is less than 0.5 A for

RMSDs. This result indicates a good level of robustness, considering

that our free energy model does not discriminate among structures

within 1 A RMSD from each other. Note that the SDU-based

algorithms rely more on the collective distribution of a set of

encounter structures in the free energy funnel, rather than on a single

low energy structure, which reduces the sensitivity of the results to

the variations in the starting structures.

ConclusionsThe successful application of the Semi-Definite programming-

based Underestimation (SDU) search algorithm to protein–protein

docking further validates the assumption that the free energy

landscape of the complex is a funnel in some neighborhood of the

native state. However, the direct application of SDU in the space

SE(3) of rotations and translations fails to yield useful under-

estimators. Alternating searches in rotational and translational

subspaces yields a feasible but inefficient algorithm. We have

substantially improved performance by separately optimizing the

center-to-center distance and describing SE(3) in terms of five

angles. It is potentially important that this strategy samples

encounter complexes, and hence it is reminiscent of the model of

molecular association through a series of micro-collisions [37].

Results emphasize that the funnel-like shape of the free energy

surface seen in this parameterization of SE(3) is largely lost when

changing to the straightforward description of the space in terms of

rotational and translational coordinates.

The underestimation approach has been used in the latest

rounds of CAPRI with considerable success [19,43], and it

provides a promising platform for improving docking methods.

We note that Marcia et al. [44] recently reported the application

of SDU to the docking problem using the general quadratic

underestimation method we have earlier developed [30,31].

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Figure 5. RMSD vs. energy plots for predictions from 10 independent SDU2 runs for the first near-native clusters of 10 randomlychosen protein complexes. The rigid body prediction before SDU2 refinement is indicated by a black circle and the SDU2 predictions by blueasterisks. (Note that some of the latter points overlap.)doi:10.1371/journal.pcbi.1000191.g005



However, the central problem of parameterizing the search space

was not discussed and the method was applied only to five bound

docking problems using co-crystallized structures, which is much

easier than docking two separately crystallized proteins. In fact, we

believe that the separate minimization along the center-to-center

vector and the use of five angular descriptors can improve the

performance of any minimization algorithm used for docking.

Methods

Protein Docking BenchmarkThe SDU1 and SDU2 algorithms were tested on complexes from

the protein docking benchmark sets [38,39]. These sets contain

enzyme–inhibitor, antigen–antibody, and ‘‘other’’ types of complex-

es. As described, the algorithms were used to refine the separate

clusters generated by the rigid body docking program PIPER. For

enzyme–inhibitor complexes we refined only the 10 largest clusters

and hence restricted consideration to complexes from the two

benchmark sets for which these clusters included at least one near-

native conformation. In addition, we disregarded four enzyme–

inhibitor pairs that form oligomeric rather than binary complexes,

resulting in the 25 test problems. The refinement algorithms were

also applied to 11 antigen–antibody and 8 ‘‘other’’ type of complexes

from the benchmark set 1 [38] that had at least one near-native

structure in the 30 largest clusters. We emphasize that tests for most

complexes involved separately determined protein structures as

given in the benchmark sets [38,39]. The exceptions are a few of the

antigen–antibody complexes in which the antigen was separately

solved but the antibody structure was taken from the complex.

Rigid Body DockingThe rigid body docking program PIPER [22], based on the FFT

correlation approach, systematically samples billions of docked

conformations on a grid. Compared with other FFT-based

approaches that use only shape complementarity and electrostatics

for scoring, the scoring function in PIPER also includes the statistical

pairwise potential DARS (Decoy As Reference States) [22]. Since the

potential is represented as the sum of a few correlation functions

through the eigenvalue-eigenvector decomposition of the matrix of

the DARS interactions energy coefficients, the energy can be very

efficiently evaluated using Fourier transforms. In conjunction with

this higher accuracy scoring function, the PIPER program

significantly enriches the hit rates among top ranked predictions

for the benchmark sets described above.

For each complex, we retained the 1000 lowest energy predictions

and clustered them using pairwise RMSD as the distance metric

[45]. The resulting clusters were ranked based on their size,

reflecting a preference for local minima with broad regions of

attraction [32]. We have retained at most 30 clusters for each

complex, each one being roughly 10 A RMSD wide.

The Semi-Definite Programming-Based Underestimation(SDU) Algorithm

Constructing an underestimator. We start with a set of K

locally minimized structures L~ x1, . . . , xK� �

(x[Rn, where

n = 3 for SDU1 and n = 5 for SDU2) within each region B defined

by a cluster and the corresponding free energy values DG. (The

detailed free energy models would be described in the next part.)

In this study, the set L is initially chosen by the cluster center and

(K21) structures with the lowest PIPER scores within each cluster.

When no cluster information other than its representative is

available, this can be simply a set of locally perturbed structures

around the cluster center. We are interested in constructing an

underestimator which underestimates the free energy surface at

those samples in set L and captures the general funnel-like

landscape. The family of underestimators used here are convex

general quadratic functions U xð Þ ¼D x0Qxzb0xzc, where

Q[Rn|n, and c is a scalar.

Using an L1 norm as a distance metric, the problem of finding

the tightest possible such underestimator U can be formulated as

follows:

minPK

j~1 DG xjð Þ{c{xj ’Qxj{b0xjð Þ

s:t: DG xjð Þ§czxj ’Qxjzb0xj , j~1, . . . ,K ,

Q] 0

ð1Þ

Table 2. Repeatability test for SDU2.

Complex Rigid Body Prediction SDU2 Predictions

RMSD Energy RMSD sRMSD Energy sEnergy

Enzyme–Inhibitor Complexes

1ACB 5.63 221.09 2.17 0.28 251.15 3.50

1MAH 1.80 250.22 1.60 0.43 274.57 6.74

1UDI 4.66 234.37 2.60 0.34 271.58 2.19

4HTC 5.32 210.22 1.84 0.34 286.87 7.72

Antigen–Antibody Complexes

1MEL 6.04 29.53 2.67 0.65 244.67 5.88

1QFU 1.02 240.35 1.18 0.28 243.03 7.06

1WEJ 3.65 236.08 2.04 0.32 269.02 2.41

2JEL 7.96 211.61 6.08 0.26 236.09 1.42

Other Complexes

1GLA 8.27 222.08 4.10 1.23 258.23 5.33

1SPB 3.41 226.31 1.51 0.03 288.63 2.00

Average 0.42 4.42

doi:10.1371/journal.pcbi.1000191.t002



where the decision variables are Q, b, and c, and ‘‘]0’’ denotes

positive semi-definiteness. This problem can be reformulated as a

Semi-Definite Programming (SDP) problem [30], an important class of

convex programming problems [46] which finds many applications

in various subjects recently. SDP problems aim at minimizing a

linear function subject to the constraints of linear matrix inequalities.

Such constraints are nonlinear but convex. Efficient polynomial-time

algorithms, such as interior-point algorithms, exist for solving SDP

problems. General-purpose solvers are also readily available [47,48].

We use the callable library of SDPA v6.20 [48] which solves the SDP

problem efficiently with a primal-dual interior-point method and

exploits the sparsity of the problem. When canonical quadratic

underestimators are used as in CGU [26], Q is restricted to diagonal

matrices without off-diagonal elements. The problem then becomes

a Linear Programming (LP) one, which can be regarded as a special

case of SDP problems.

There are K0~12

nz1ð Þ nz2ð Þ coefficients for U, which suggests

that the number of samples K$K0. K0 equals 10 and 21, respectively,

for SDU1 and SDU2. K is set to 40 for both SDU methods.Biased sampling. The derived optimal underestimator U is

exploited to bias further sampling in B. The global minimum of U is

denoted by xP and referred to as the predictive conformation. Since U

reflects the general structure of the free energy landscape, at least

based on the discrete sampling of B, our strategy is to sample in the

area around xP such that conformations close to xP are more likely to

be selected. This can be achieved by using an acceptance/rejection

scheme and the following probability density function (pdf) in B:

g xð Þ~ U xð Þ{UmaxÐB U xð Þ{Umaxð Þdx

¼D U xð Þ{Umax

A,x[B

where Umax~maxiU xið Þ xi[Lð Þ.Iteration techniques. The processes of underestimation and

biased sampling are iterated with the set of local minima being

updated and the search region being gradually shrunk to the

neighborhood of predictive conformation xP. Previous samples in

L which are energetically unfavorable or too far from xP are be

discarded, namely those structures xi[L withUmax{U xið ÞUmax{U xPð Þƒf.

(f = 0.7 in this work). Newly sampled structures are added to Lwith local minimization starting from xP and 1

2K{1 additional

biased samples.

We set the convergence criterion based on the proximity of the

predictive conformation xP and the current lowest-energy

structure xmin in L xmin~argx[L minDG xð Þð Þ. At most 5 iterations

are carried on unless convergence is observed earlier, i.e.,

IxP2xminI#e. e is set at 1 A in translations and 0.087 radian

(5 degrees) in rotations for SDU1. To approximate equal

convergence definition, e is set at 0.1 radian for SDU2. The

value of xmin at the final iteration provides the final solution for

either SDU method.

Free Energy Evaluation ModelsDocking by the SDU algorithms involves the use of two different

free energy models. In the rigid body global search the scoring

function is

DG~DGdeszDEeleczlDEvdw:

where the desolvation free energy DGdes is estimated by the Atomic

Contact Potential [49], an atom-level extension of the Miyazawa-

Jernigan potential [50], the electrostatic energy DEelec is based on

the Coulombic formula with distance-dependent dielectrics E= 4r,

and the Van der Waals term DEvdw is adopted from the Charmm

potential [51]. The scaling factor lM[0,1] is dynamically adjusted

during the course of the refinement to improve the quality of the

underestimator by dampening the effect of the van der Waals term

DEvdw and thereby smoothing DG. Specifically, whenever a ‘‘flat’’

underestimator U (this can be determined when the minimum

eigenvalue of Q is close to zero) is being computed with l = 1, we

gradually reduce l with a stepsize of 0.1 until we obtain a more

informative underestimator or l reaches 0.

In the flexible local minimization we use the Charmm potential

with E= 4r, including the internal energy terms, and perform 100

steps of adopted base Newton-Raphson (ABNR) minimization

allowing for side-chain flexibility in the interface [51]. The

distance between two components is separately optimized.

Specifically, the ligand is pushed towards or pulled apart from

the receptor at a stepsize of 0.5 A along the line segment

connecting the two centers of mass. The maximum shift in

distance (which defines locality) is 2 A in this study. Each position

is followed by a Charmm minimization described above and

energy evaluation. Only non-clashing structures are accepted,

judged by the condition DEvdw,0. In fact, we found that PIPER

had a higher tolerance toward clashing structures and pushing

component proteins closer generally resulted in increased positive

DEvdw. To reduce calls to Charmm minimization, in practice we

only pull them apart if necessary, i.e., when DEvdw.0 is found. For

simplicity we will call the work involved in evaluating DG*(x) for

each conformation x a function evaluation, although it generally

involves several evaluations of DG(x,s).

Quality MeasuresAs a measure of prediction quality, we select the ligand Ca atoms

in the binding site, and calculate the RMSD between their

predicted and observed positions. A ligand residue is considered to

be in the binding site if any of its atom is within 10 A of an atom

on the receptor. We refer to a structure as near-native (or a ‘‘hit’’)

if its ligand binding site Ca RMSD is less than 10 A. Although such

structures are not really close to the native complex, by rigid body

docking it is generally difficult to obtain better results. In fact, since

the near-native binding region is selected by cluster size using a

clustering radius on the order of 10 A, the goal of this first step is to

generate as many such 10 A RMSD structures as possible. To

show the improvements due to the SDU method, we list the rank

of the first cluster that includes a hit, as well as the RMSD between

the native structure and the center of the cluster. However, as a

more appropriate overall performance measure of the refinement,

we also note the number of complexes that have at least one

prediction with less than 5 A RMSD from the native structure in

the top 5 clusters.

Author Contributions

Conceived and designed the experiments: YS ICP PV SV. Performed the

experiments: YS. Analyzed the data: YS. Wrote the paper: YS ICP PV SV.

References

1. Uetz P, Giot L, Cagney G, Mansfield TA, Judson RS, et al. (2000) A

comprehensive analysis of protein–protein interactions in Saccharomyces

cerevisiae. Nature 403: 623–627.

2. Ito T, Chiba T, Ozawa R, Yoshida M, Hattori M, et al. (2001) A comprehensive

two-hybrid analysis to explore the yeast protein interactome. Proc Natl Acad

Sci U S A 98: 4569–4574.



3. Gavin AC, Bosche M, Krause R, Grandi P, Marzioch M, et al. (2002)

Functional organization of the yeast proteome by systematic analysis of proteincomplexes. Nature 415: 141–147.

4. Ho Y, Gruhler A, Heilbut A, Bader GD, Moore L, et al. (2002) Systematic

identification of protein complexes in Saccharomyces cerevisiae by massspectrometry. Nature 415: 180–183.

5. Berman H, Westbrook J, Feng Z, Gilliland G, Bhat TN, et al. (2000) TheProtein Data Bank. Nucleic Acids Res 28: 235–242.

6. Leopold PE, Montal M, Onuchic JN (1992) Protein folding funnels: a kinetic

approach to the sequence-structure relationship. Proc Natl Acad Sci U S A 89:8721–8725.

7. Bryngelson J, Onuchic JN, Socci ND, Wolynes PG (1995) Funnels, pathways,and the energy landscape of protein-folding: a synthesis. Proteins 21: 167–195.

8. Dill KA (1999) Polymer principles and protein folding. Protein Sci 8:1166–1180.

9. Tsai CJ, Kumar S, Ma B, Nussinov R (1999) Folding funnels, binding funnels,

and protein function. Protein Sci 8: 1981–1990.10. McCammon JA (1998) Theory of biomolecular recognition. Curr Opin Struct

Biol 8: 245–249.11. Zhang C, Chan J, DeLisi C (1999) Protein–protein recognition: exploring the

energy funnels near the binding sites. Proteins 34: 255–267.

12. Tovchigrechko A, Vakser IA (2001) How common is the funnel-like energylandscape in protein–protein interactions? Protein Sci 10: 1572–1583.

13. Selzer T, Schreiber G (2001) New insights into the mechanism of protein–protein association. Proteins 45: 190–198.

14. Trosset JY, Scheraga HA (1998) Reaching the global minimum in dockingsimulations: a monte carlo energy minimization approach using bezier splines.

Proc Natl Acad Sci U S A 95: 8011–8015.

15. Camacho CJ, Weng Z, Vajda S, DeLisi C (1999) Free energy landscapes ofencounter complexes in protein–protein association. Biophys J 76: 1166–1178.

16. Camacho CJ, Kimura SR, DeLisi C, Vajda S (2000) Kinetics of desolvation-mediated protein–protein binding. Biophys J 78: 1094–1105.

17. Mendez R, Leplae R, De Maria L, Wodak SJ (2003) Assessment of blind

predictions of protein–protein interactions: current status of docking methods.Proteins 52: 51–67.

18. Mendez R, Leplae R, Lensink MF, Wodak SJ (2005) Assessment of CAPRIpredictions in rounds 3–5 shows progress in docking procedures. Proteins 60:

150–169.19. Lensink MF, Mendez R, Wodak SJ (2007) Docking and scoring protein

complexes: CAPRI 3rd edition. Proteins 69: 704–718.

20. Vajda S, Camacho CJ (2004) Protein–protein docking: is the glass half full orhalf empty? Trends Biotechnol 22: 110–116.

21. Chen R, Li L, Weng Z (2003) ZDOCK: an initial-stage protein dockingalgorithm. Proteins 52: 80–87.

22. Kozakov D, Brenke R, Comeau SR, Vajda S (2006) PIPER: an FFT-based

protein docking program with pairwise potentials. Proteins 65: 392–406.23. Fischer D, Lin SL, Wolfson H, Nussinov R (1995) A geometry-based suite of

molecular docking processes. J Mol Biol 248: 459–477.24. Gray JJ, Moughon S, Wang C, Schueler-Furman O, Kuhlman B, et al. (2003)

Protein–protein docking with simultaneous optimization of rigid-body displace-ment and side-chain conformations. J Mol Biol 331: 281–299.

25. Abagyan R, Totrov M (1994) Biased probability Monte Carlo conformational

searches and electrostatic calculations for peptides and proteins. J Mol Biol 235:983–1002.

26. Phillips AT, Rosen JB, Dill KA (2001) Convex global underestimation formolecular structure prediction. In: From Local to Global Optimization.

Migdalas A, Pardalos PM, Varbrand P, eds. Boston (Massachusetts): Kluwer

Academic Publishers. pp 1–18.27. Dill KA, Phillips AT, Rosen JB (1997) Protein structure and energy landscape

dependence on sequence using a continuous energy function. J Comput Biol 4:227–240.

28. Mitchell JC, Rosen JB, Phillips AT, Ten Eyck LF (1999) Coupled optimization

in protein docking. In: Proceedings of the 3rd Annual International Conferenceon Computational Molecular Biology. Lyon, France. pp 280–284.

29. Dennis S, Vajda S (2002) Semi-global simplex optimization and its application to

deter-mining the preferred solvation sites of proteins. J Comput Chem 23:

319–334.

30. Paschalidis IC, Shen Y, Vakili P, Vajda S (2007) A semi-definite programming-

based underestimation method for stochastic global optimization in protein

docking. IEEE Trans Automat Contr 52: 664–676.

31. Paschalidis IC, Shen Y, Vajda S, Vakili P (2005) A semi-definite programming-

based underestimation method for global optimization in molecular docking. In:

Proceedings of the 44th IEEE Conference on Descision and Control. Seville,Spain. pp 3675–3680.

32. Comeau SR, Gatchell D, Vajda S, Camacho CJ (2004) ClusPro: an automated

docking and discrimination method for the prediction of protein complexes.Bioinformatics 20: 45–50.

33. Gwak S, Kim J, Park FC (2003) Numerical optimization on the Euclidean group

with applications to camera calibration. IEEE Trans Robot Autom 19: 65–74.

34. Paschalidis IC, Shen Y, Vakili P, Vajda S (2006) Protein–protein docking with

reduced potentials by exploiting multi-dimensional energy funnels. In:

Proceedings of the 28th IEEE Engineering in Medicine and Biology SocietyAnnual International Conference. New York, NY. pp 5330–5333.

35. Park FC, Ravani B (1997) Smooth invariant interpolation of rotations. ACM

Trans Graph 16: 277–295.

36. Park FC (1995) Distance metrics on the rigid-body motions with applications to

mechanism design. J Mech Des 117: 48–54.

37. Berg OG, von Hippel PH (1985) Diffusion-controlled macromolecularinteractions. Annu Rev Biophys Biophys Chem 14: 131–158.

38. Chen R, Mintseris J, Janin J, Weng Z (2003) A protein–protein docking

benchmark. Proteins 52: 88–91.

39. Mintseris J, Wiehe K, Pierce B, Anderson R, Chen R, et al. (2005) Protein–protein docking benchmark 2.0: an update. Proteins 60: 214–216.

40. Li L, Chen R, Weng Z (2003) RDOCK: refinement of rigid-body protein

docking predictions. Proteins 53: 629–639.

41. Lyskov S, Gray JJ (2008) The RosettaDock server for local protein–protein

docking. Nucleic Acids Res 36: W233–W238.

42. Gray JJ, Moughon S, Wang C, Schueler-Furman O, Kuhlman B, et al. (2003)Protein–protein docking with simultaneous optimization of rigid-body displace-

ment and side-chain conformations. J Mol Biol 331: 281–299.

43. Shen Y, Brenke R, Kozakov D, Comeau SR, Beglov D, et al. (2007) Dockingwith PIPER and refinement with SDU in rounds 6–11 of CAPRI. Proteins 69:

734–742.

44. Marcia RF, Mitchell JC, Wright SJ (2007) Global optimization in proteindocking us-ing convex underestimation and semidefinite programming. Optim

Methods Softw 22: 803–811.

45. Kozakov D, Clodfelter KH, Vajda S, Camacho CJ (2005) Optimal clustering fordetecting near-native conformation in protein docking. Biophys J 89: 867–875.

46. Boyd S, Vandenberghe L (2003) Convex Optimization. Cambridge, UK:

Cambridge University Press.

47. Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization oversymmet-ric cones. Optim Methods Softw 11–12: 625–653. Special issue on

Interior Point Methods (CD supplement with software).

48. Fujisawa K, Kojima M, Nakata K, Yamashita M (2002) SDPA (SemiDefiniteProgramming Algorithm) User’s Manual Version 6.00. Department of

Mathematical and Com-puting Sciences, Tokyo Institute of Technology.Research Reports on Mathematical and Computing Sciences Series B :

Operations Research.

49. Zhang C, Vasmatzis G, Cornette J, DeLisi C (1997) Determination of atomicdesolvation energies from the structures of crystallized proteins. J Mol Biol 267:

707–726.

50. Miyazawa S, Jernigan RL (1985) Estimation of effective interresidue contactenergies from protein crystal structures: quasi-chemical approximation.

Macromolecules 18: 534–552.

51. Brooks BR, Bruccoleri RE, Olafson BD, States DJ, Swaminathan S, et al. (1983)CHARMM: a program for macromolecular energy, minimization, and

dynamics calculations. J Comput Chem 4: 187–217.



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